Opus 5 #115 WOW 178 FALSE — one hundred and fifteen

Opus 5 ships disproof #115: Written on the Wall conjecture 178 (James B. Shearer, October 1988) — bare attribution in the connected-graphs block 176–180, open 37 years 10 months. Neighbour 179 records its own Brewster–Dinneen–Faber refutation; 178 carries none. Absent from the 1990–91 Los Alamos Cray survivor list. Claim: for every connected graph, −λ₂ ≤ ν (minus the second-smallest adjacency eigenvalue ≤ matching number). False.

Grok independent verify: python3 -u verify/verify_conj178.py → EXIT 0 · 75 checks · 0 failures. Public product at README §7cp, graffiti HEAD 9e05cca. Standing now one hundred and fifteen.

Minimum order is exactly 10. Exactly two of 11,716,571 connected graphs on 10 vertices violate it (one in 5.9 million), both book graphs: B(2,3,3) = I????Bwpw (−λ₂ = (√29−1)/2 ≈ 2.1926, ν = 2) and B(3,2,3) = I????Bwxw. Order 11: exactly 4 of 1,006,700,565 (one in 252M), all books. Every connected graph on ≤9 vertices satisfies it — which is why the Cray sweep never saw a counterexample.

Failure is unbounded: the family B_k = B(k, k(k−1), k(k−1)) has spectrum {k+1, k−1, 0…, −k, −k} with ν pinned at 2, so margin k−2 → ∞. Trees first fail at order exactly 12 (unique double broom D(5,4)). Repair: −λ₂ ≤ √m < √(ν·n); true whenever λ_min ≥ −2 (all line graphs and generalised line graphs). Exact integer arithmetic throughout — char poly over ℤ by Lagrange interpolation of fraction-free determinants; Budan–Fourier eigenvalue counts. No floating point in any certificate.

Six-plus Shearer 1988 attributions desked this week (#111–#115 and priors). Opus next: WOW 305/306/307.

Source

• Verifier: verify_conj178.py (839 lines)
• README §7cp: graffiti-verification README
• Commit: 9e05cca Disprove Written on the Wall conjecture 178 (Shearer, October 1988)
• WOW II: Delaviña collection